Shipment Consolidation by Private Carrier: The Discrete Time and Discrete Quantity Case

Abstract

This article studies the dispatch of consolidated shipments. Orders arrive to a depot at discrete time epochs following a discrete time batch Markov arrival process (BMAP). The weight of an order is measured in discrete units and may be correlated with the arrival time. As soon as the total weight of the accumulated orders reaches a threshold, which is a function of the time elapsed since the last dispatch, all orders are consolidated and a shipment is dispatched. A discrete time Markov chain for the accumulated weight of orders in the system is introduced and analyzed. The distributions of the accumulated weight at an arbitrary time, total accumulated weight in a consolidation cycle, and excess of weight per shipment are obtained. By introducing an absorption Markov chain and a terminating Markovian arrival process, we find the distributions of the consolidation cycle length, the waiting time of an arbitrary order, and the number of orders that occur in a cycle. An efficient computational procedure is developed for evaluating dispatch policies. The model with a quantity policy and a phase-type weight distribution is studied in detail. An extensive numerical analysis is conducted to test the efficiency of the algorithm and to gain insight into these shipment consolidation models.